Edexcel S2 (Statistics 2) 2016 January

1. The manager of a clothing shop wishes to investigate how satisfied customers are with the quality of service they receive. A database of the shop's customers is used as a sampling frame for this investigation.
   1. Identify one potential problem with this sampling frame.
   Customers are asked to complete a survey about the quality of service they receive. Past information shows that \(35 \%\) of customers complete the survey. A random sample of 20 customers is taken.
2. Write down a suitable distribution to model the number of customers in this sample that complete the survey.
3. Find the probability that more than half of the customers in the sample complete the survey.

1. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) Given that \(\mathrm { P } ( 3 < X < 5 ) = \frac { 1 } { 8 }\) and \(\mathrm { E } ( X ) = 4\)
   1. find the value of \(a\) and the value of \(b\)
   2. find the value of the constant, \(c\), such that \(\mathrm { E } ( c X - 2 ) = 0\)
   3. find the exact value of \(\mathrm { E } \left( X ^ { 2 } \right)\)
   4. find \(\mathrm { P } ( 2 X - b > a )\)
      
   5. Left-handed people make up \(10 \%\) of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.
   6. 1. Write down an expression for the exact value of \(\mathrm { P } ( Y \leqslant 1 )\)
      2. Evaluate your expression, giving your answer to 3 significant figures.
   7. Using a Poisson approximation, estimate \(\mathrm { P } ( Y \leqslant 1 )\)
   8. Using a normal approximation, estimate \(\mathrm { P } ( Y \leqslant 1 )\)
   9. Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm { P } ( Y \leqslant 1 )\)

4. A continuous random variable \(X\) has cumulative distribution function $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 0
\frac { 1 } { 4 } x & 0 \leqslant x \leqslant 1
\frac { 1 } { 20 } x ^ { 4 } + \frac { 1 } { 5 } & 1 < x \leqslant d
1 & x > d \end{array} \right.$$

1. Show that \(d = 2\)
2. Find \(\mathrm { P } ( X < 1.5 )\)
3. Write down the value of the lower quartile of \(X\)
4. Find the median of \(X\)
5. Find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( X > 1.9 ) = \mathrm { P } ( X < k )\)

5. The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1

1. Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods.
2. Find the probability that this volcano does not erupt in a randomly selected 20 year period. The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.
3. Use the tables to find the value of \(w\) A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1 She selects a 100 year period at random in order to test her claim.
4. State the null hypothesis for this test.
5. Determine the critical region for the test at the \(5 \%\) level of significance.
   

6. A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x ^ { 2 } + b x & 1 \leqslant x \leqslant 7
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(114 a + 24 b = 1\) Given that \(a = \frac { 1 } { 90 }\)
2. use algebraic integration to find \(\mathrm { E } ( X )\)
3. find the cumulative distribution function of \(X\), specifying it for all values of \(x\)
4. find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
5. use your answer to part (d) to describe the skewness of the distribution.
   

1. A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution.

The fisherman takes 5 fishing trips each lasting 1 hour.

1. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour. Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,
2. carry out the test at the \(5 \%\) level of significance. State your hypotheses clearly.
   (6) by the fisherman in an hour follows a Poisson distribution.
   The fisherman takes 5 fishing trips each lasting 1 hour.
3. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips.
   7.
   
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